Geometry, MAT 631, Fall 2009
Elementary geometry from an advanced standpoint.
This course is part of the
Mathematics Teacher Transformation Institute, an
NSF Math & Science Partnership program.
|Geometry, MAT631, Fall 2009
Grading Policy: 11 in class projects, no exams.
Homework: read over material, and finish projects.
Prerequisites: participation in
|Professor C. Sormani
Office: Gillet Hall 200B
Email: sormanic (at) member.ams.org
Classical Euclidean Geometry using postulates and axioms
will be combined with a more modern transformation perspective
in this course tailored to provide teachers with an in depth
understanding of the geometry in New York State's
new high school
Text: (to be provided by MTTI)
College Geometry: A discovery approach, by David C. Kay, 2nd Edition.
Other Supplies: (not provided, bring your own)
Grades: The grade will be based on the projects. Projects
must be completed with two column proofs and neat hand drawn sketches
using straight edges and compasses. Proofs must be done in two columns:
Statements and Justifications. The projects will be graded harshly with
deductions taken for errors. However, the lowest grades will be dropped
and lots of extra credit will be assigned. Projects will be done in class
and submitted the following week at the beginning of class. Most projects
will include at least one very difficult problems.
Appendix B has a review of high school geometry.
Appendix C discusses Geometers Sketchpad (we won't cover).
Appendix F has a list of axioms for quick reference (bring to class).
This book approaches geometry using the three classical
geometries: Euclidean, spherical and hyperbolic. In this course
we focus on the Euclidean part as this directly relates to the
NYS curriculum, particular the Geometry Regents Exam which includes
The projects in the following syllabus may be changed or reordered
as we would like the projects to be relevant to the participants. For
the proofs we will be following Kay's set of axioms, which correspond more
closely to the NYS curriculum and modern mathematics
than Euclid's ancient set of axioms.
Sept 9: Initial Meeting:
Required Incoming Assessment (necessary for statistical evaluation).
Sept 16: First Axioms
Extra Credit: Early History of pi (16-17) and Archimedes' Calculation
2.3-2.4 Points, Lines, Planes, Distances, (read pp 70-73, pp77-82)
Sept 23: Measuring Lengths and Angles and Convex Sets:
Circles, Rays, Angles, Right Angles, Perpendicular Lines
Project 1: Write axioms and theorems
with proofs assigned in class in order
and drawings of each axiom and theorem.
2.4-2.6 Ruler Postulate, Protractor Postulate,
Convex Sets and Crossbar Theorem (read pp 83-85, 90-99, 103-110)
Sept 30: Congruent Triangles:
Project 2: Read pp 83-85 & 90-99 then prove (1) Angle Construction
and (2) Unique Perpendicular Line Theorem. Read pp 103-104 then
prove (3) Intersection of Convex sets is Convex but not unions.
Read pp105-106 and prove (4)Thm 1. Read p 107 and prove (5) Pasch's Thm.
Read pp 108-109 and prove (6) Thm 3 and (7) Crossbar Thm. All proofs
must be written in two columns. Some proofs may be found in the book
or on the web. You must understand what you have written and be
able to present it.
3.1, 3.3-3.4: SAS, ASA, etc
Oct 7: Concurrent Lines
Oct 14: Parallel Postulate:
Project 3: Handout in class. Otherwise do the following:
(1) Read handout and fill in justifications for
the proof of SSS.
Read 123-124 and 139-142 on SAS Postulate/Hypothesis and
ASA Theorem, Ex 1-3, Isosceles Triangle defn and Theorem,
Construction of the perpendicular bisector.
(2) On graph paper using compass and straight edge, construct the
perpendicular bisector of segment AB, where A=(1,3) and B=(5,7), then
where A=(-2,4) and B=(-6, 2).
(3) Suppose A,B, and C are noncolinear, prove there exists a unique line
perpendicular to line BC passing through A. Hint: Find a point D such that
BD=AD and measure angle CBA= measure and angle CBD.
(4) Angle bisector construction: Given angle ABC with AB=BC,
show that if a point X has AX=CX, then ray BX is the angle bisector.
You may use Theorem SSS to do this proof.
(5) Write up proof of SSA as stated on page 175 (or in handout).
Read proof of ASA in handout.
Note we pospone the proof of AAS which can be proven quickly
once we know the sum of the angles of a triangle is 180 degrees.
That cannot be proven until we introduce Euclid's Parallel Postulate.
Right now everything we've proven is true for more general geometries
like the sphere and hyperbolic space. For a proof of AAS in these
more general spaces, you may read 3.6 if you wish.
Extra Credit: Resubmit Pythagoras proofs
adding in angle justifications.
Parallel Lines, Angle Sum Theorem, Parallelograms 4.1-4.2
Oct 21: Parallelograms:
Parallel Postulate Project pages 1-3
Extra Credit: Solid Geometry: Read 7.1, write out two column
proofs of Theorems 1-3, Relate to
the NYS Standards Concepts G.G.1-G.G.9,
Parallelograms, rhombuses and squares
Oct 28: Similar triangles:
Nov 4: Pythagoras and its Converse:
Nov 11: Symmetries and Isometries:
Nov 18: Circles, Arcs and Chords:
Lesson begins with examples of bridges, why are some rigid
while others collapse and why equilateral triangles make straight lines.
Parallel Postulate Project page 4
Circles 3.8 and 4.5, NYS Standards GG49-GG53.
Nov 25: Coordinate Geometry and Vectors:
Project 10: Circles Project
pages 1-3. Page 4 is Extra Credit.
4.3, 4.7 Coordinate Plane, Quadrants, Lines through the Origin,
Dec 2: Transformations, Areas :
Coordinate Plane Project
Shift Isometry and Lines, Skews/Shears and Dilations,
and Area 4.6, Skews and Dilations 5.1,
Dec 9: Solid Geometry
Complete both projects as project 12.
7.1-7.4 p 297
Dec 16: Discussion of Direct Application in Classroom:
Project 13: Intuitive Solid Geometry Project
Bring NYS Standards and Questions